Quadratic Sequence nth Term Calculator
The nth-term formula an² + bn + c of a quadratic sequence from its first three terms, with any term, the next term and the sum of the first n.
Results
The dashed line marks the value you entered.
What this tool does
A sequence is quadratic when the differences between neighbouring terms are not constant but the differences of those differences are — the second difference. Half the second difference is the coefficient of n². The method used in school is the same: subtract an² from every term and what is left is a linear sequence bn + c. For 3, 8, 15, 24 the first differences are 5, 7, 9, the second difference is 2, so a = 1 and the nth term is n² + 2n. Three terms always fit exactly one quadratic, but they do not prove the pattern continues: check the fourth term if you have it.
Formula
second difference = t₃ − 2t₂ + t₁ · a = second difference ÷ 2 · b = (t₂ − t₁) − 3a · c = t₁ − a − b · tₙ = a n² + b n + c
Variables
| Symbol | Meaning | Unit |
|---|---|---|
t1 | First term | — |
t2 | Second term | — |
t3 | Third term | — |
n | Which term you want (n) | — |
T | The nth term | — |
A | a — coefficient of n² | — |
B | b — coefficient of n | — |
C | c — constant | — |
D1 | First difference (second − first term) | — |
D2 | Second difference (constant) | — |
NX | Next term (the 4th) | — |
SM | Sum of the first n terms | — |
Worked example
- First term3
- Second term8
- Third term15
- Which term you want (n)10
- The nth term120.0000
- a — coefficient of n²1.000000
- b — coefficient of n2.000000
- c — constant0.000000
- First difference (second − first term)5.0000
- Second difference (constant)2.0000
- Next term (the 4th)24.0000
- Sum of the first n terms495.0000
Limitations
- The calculation runs at full precision and only the display is rounded. If you copy an intermediate value and retype it, small differences can appear.
- The default values are typical reference figures, not measurements of your situation. Replace them with your own data whenever you have it.